Three-dimensional rotations. More...

#include <Rotation.h>

Inheritance diagram for Rotation:

[legend]

Public Member Functions
double	GetAlpha (void)
	Returns the current value of \(\alpha\) [rad]. More...

double	GetBeta (void)
	Returns the current value of \(\beta\) [rad]. More...

double	GetCosAlpha (void)
	Returns the current value of \(\cos(\alpha)\). More...

double	GetCosBeta (void)
	Returns the current value of \(\cos(\beta)\). More...

double	GetCosGamma (void)
	Returns the current value of \(\cos(\gamma)\). More...

double	GetGamma (void)
	Returns the current value of \(\gamma\) [rad]. More...

double	GetOmega (void)
	Returns the current value of \(\omega\). More...

double	GetPsi (void)
	Returns the current value of \(\Psi\). More...

double	GetRotationAngle (void)
	Returns the rotation angle from \(\cal{R}^{\prime}\) to \(\cal{R}\) [rad]. More...

double	GetRotationAxisXp (void)
	Returns the \(x^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\). More...

double	GetRotationAxisYp (void)
	Returns the \(y^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\). More...

double	GetRotationAxisZp (void)
	Returns the \(z^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\). More...

double	GetRxx ()
	Returns the rotation matrix element \(R_{xx}\). More...

double	GetRxy ()
	Returns the rotation matrix element \(R_{xy}\). More...

double	GetRxz ()
	Returns the rotation matrix element \(R_{xz}\). More...

double	GetRyx ()
	Returns the rotation matrix element \(R_{yx}\). More...

double	GetRyy ()
	Returns the rotation matrix element \(R_{yy}\). More...

double	GetRyz ()
	Returns the rotation matrix element \(R_{yz}\). More...

double	GetRzx ()
	Returns the rotation matrix element \(R_{zx}\). More...

double	GetRzy ()
	Returns the rotation matrix element \(R_{zy}\). More...

double	GetRzz ()
	Returns the rotation matrix element \(R_{zz}\). More...

double	GetSinAlpha (void)
	Returns the current value of \(\sin(\alpha)\). More...

double	GetSinBeta (void)
	Returns the current value of \(\sin(\beta)\). More...

double	GetSinGamma (void)
	Returns the current value of \(\sin(\gamma)\). More...

double	GetUpx (const double aThetaPrime, const double aPhiPrime)
	Returns the \(u^{\prime}_x\) coordinate in the \(\cal{R}^{\prime}\) reference frame. More...

double	GetUpy (const double aThetaPrime, const double aPhiPrime)
	Returns the \(u^{\prime}_y\) coordinate in the \(\cal{R}^{\prime}\) reference frame. More...

double	GetUpz (const double aThetaPrime)
	Returns the \(u^{\prime}_z\) coordinate in the \(\cal{R}^{\prime}\) reference frame. More...

double	GetUx (const double aTheta)
	Returns the \(u_x\) coordinate in the \(\cal{R}\) reference frame. More...

double	GetUy (const double aTheta, const double aPhi)
	Returns the \(u_y\) coordinate in the \(\cal{R}\) reference frame. More...

double	GetUz (const double aTheta, const double aPhi)
	Returns the \(u_z\) coordinate in the \(\cal{R}\) reference frame. More...

void	SetAlpha (const double aAlpha=0.0)
	Sets a new \(\alpha\) value. More...

void	SetAlphaBetaGamma (const double aAlpha, const double aBeta, const double aGamma)
	Sets Euler angles, \(\alpha, \beta, \gamma\), for the rotation. More...

void	SetBeta (const double aBeta=0.0)
	Sets a new \(\beta\) value. More...

void	SetGamma (const double aGamma=0.0)
	Sets a new \(\gamma\) value. More...

void	SetUnitQuaternion (const double aQ0, const double aQ1, const double aQ2, const double aQ3, const bool aRtoRprime=true)
	Sets Euler angles from a unit quaternion. More...

double	TransformGetCosPhi (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\cos\varphi\) in the reference frame \(\cal{R}\). More...

double	TransformGetCosPhiPrime (const double aTheta, const double aPhi)
	Returns the \(\cos\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetCosTheta (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\cos\theta\) in the reference frame \(\cal{R}\). More...

double	TransformGetCosThetaPrime (const double aTheta, const double aPhi)
	Returns the \(\cos\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetSinPhi (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\sin\varphi\) in the reference frame \(\cal{R}\). More...

double	TransformGetSinPhiPrime (const double aTheta, const double aPhi)
	Returns the \(\sin\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetSinTheta (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\sin\theta\) in the reference frame \(\cal{R}\). More...

double	TransformGetSinThetaPrime (const double aTheta, const double aPhi)
	Returns the \(\sin\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetTanPhi (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\tan\varphi\) in the reference frame \(\cal{R}\). More...

double	TransformGetTanPhiPrime (const double aTheta, const double aPhi)
	Returns the \(\tan\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetTanTheta (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\tan\theta\) in the reference frame \(\cal{R}\). More...

double	TransformGetTanThetaCosPhi (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\tan\theta\cos\varphi\) in the reference frame \(\cal{R}\). More...

double	TransformGetTanThetaPrime (const double aTheta, const double aPhi)
	Returns the \(\tan\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\). More...

double	TransformGetTanThetaSinPhi (const double aThetaPrime, const double aPhiPrime)
	Returns the \(\tan\theta\sin\varphi\) in the reference frame \(\cal{R}\). More...

Constructors and destructors
	Rotation (const double aAlpha, const double aBeta, const double aGamma)
	Rotation class constuctor. More...

virtual	~Rotation (void)
	Rotation class destructor. More...

Protected Attributes
double	rot_alpha
	Euler angle \(\alpha\). More...

double	rot_beta
	Euler angle \(\beta\). More...

double	rot_gamma
	Euler angle \(\gamma\). More...

Private Attributes
double	c_alpha
	\(\cos(\alpha)\). More...

double	c_beta
	\(\cos(\beta)\). More...

double	c_gamma
	\(\cos(\gamma)\). More...

double	s_alpha
	\(\sin(\alpha)\). More...

double	s_beta
	\(\sin(\beta)\). More...

double	s_gamma
	\(\sin(\gamma)\). More...

Detailed Description

Three-dimensional rotations.

Defintions

This class is designed to perform three-dimensional rotations between two reference frames. The first reference frame \(\cal{R}^{\prime}\) is defined with a coordinate system \((x^{\prime},y^{\prime},z^{\prime})\). The second reference frame \(\cal{R}\) is defined with a coordinate system \((x,y,z)\). A three-dimensional rotation is needed to change from one coordinate system to the other. This rotation is described using the Euler angles \((\alpha, \beta, \gamma)\). In the following figure, the geometrical definitions are represented. The coordinate system \((x^{\prime},y^{\prime},z^{\prime})\) is represented in black and \((x,y,z)\) is represented in blue. The \(x\)-axis orientation is given by the spherical angles \((\omega,\psi)\).

Geometrical definitions. The rotation to change the system of coordinates is given by the Euler angles and is represented in green.

Change of reference frames

To move from one coordinate system to another, a three-dimensional rotation is performed. To develop the formalism associated to such a transform, we adopt the intrinsic Euler angle rotation system. First we define the line of nodes, \(N\), as the the common perpendicular to the \(z^{\prime}\) and \(z\) axes. The three Euler angles are then represented in green in the figure above:

\(\alpha\) is the angle between the \(x^{\prime}\) axis and the \(N\) axis, \([0,2\pi[\).
\(\beta\) is the angle between the \(z^{\prime}\) axis and the \(z\) axis, \([0,\pi]\).
\(\gamma\) is the angle between the \(N\) axis and the \(x\) axis, \([0,2\pi[\).

The rotation from \(\cal{R}^{\prime}\) to \(\cal{R}\) is decomposed into a sequence of three elementary rotations that occur about the moving \(\cal{R}\) axes. Starting from a position where both reference axis systems are aligned, the first rotation is performed around the \(z^{\prime}=z\) axis with an angle equal to \(\alpha\), so that the \(x\) axis is aligned with the line of nodes. The second rotation is performed around the line of nodes (or \(x\)) by an angle \(\beta\) so that the \(z\) axis reaches its final position. The final rotation occurs about \(z\) by an angle \(\gamma\) and \(\cal{R}\) is in its final position. The rotation matrix for this transformation is ( \(c\) is used for \(\cos\) and \(s\) is used for \(\sin\)):

\[ \left( {\begin{array}{c} x \\ y \\ z \end{array} } \right) = R \left( {\begin{array}{c} x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array} } \right) = \left[ {\begin{array}{ccc} c_{\alpha}c_{\gamma}-s_{\alpha}c_{\beta}s_{\gamma} & -s_{\alpha}c_{\gamma}-c_{\alpha}c_{\beta}s_{\gamma} & s_{\beta}s_{\gamma} \\ c_{\alpha}s_{\gamma}+s_{\alpha}c_{\beta}c_{\gamma} & -s_{\alpha}s_{\gamma}+c_{\alpha}c_{\beta}c_{\gamma} & -s_{\beta}c_{\gamma} \\ s_{\alpha}s_{\beta} & c_{\alpha}s_{\beta} & c_{\beta} \end{array} } \right] \left( {\begin{array}{c} x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array} } \right) = \left[ {\begin{array}{ccc} R_{xx} & R_{xy} & R_{xz} \\ R_{yx} & R_{yy} & R_{yz} \\ R_{zx} & R_{zy} & R_{zz} \end{array} } \right] \left( {\begin{array}{c} x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array} } \right) \]

obtained by applying 3 elementary rotations \(R = R_z(\gamma) \times R_x(\beta) \times R_z(\alpha)\), with:

\[ R_z(\alpha) = \left[ {\begin{array}{ccc} \cos(\alpha) & -\sin(\alpha) & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 \\ 0 & 0 & 1 \end{array} } \right] \]

\[ R_x(\beta) = \left[ {\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\beta) & -\sin(\beta) \\ 0 & \sin(\beta) & \cos(\beta) \end{array} } \right] \]

\[ R_z(\gamma) = \left[ {\begin{array}{ccc} \cos(\gamma) & -\sin(\gamma) & 0 \\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 \end{array} } \right] \]

This rotation matrix is used to transform the system of coordinates of a given position: \(\vec{u} = R \vec{u}^{\prime}\) where:

\[ \vec{u} = \left( {\begin{array}{c} u_x \\ u_y \\ u_z \end{array} } \right) = \left( {\begin{array}{c} \cos\theta \\ \sin\theta\cos\varphi \\ \sin\theta\sin\varphi \end{array} } \right) \]

and

\[ \vec{u}^{\prime} = \left( {\begin{array}{c} u^{\prime}_x \\ u^{\prime}_y \\ u^{\prime}_z \end{array} } \right) = \left( {\begin{array}{c} \sin\theta^{\prime}\cos\varphi^{\prime} \\ \sin\theta^{\prime}\sin\varphi^{\prime} \\ \cos\theta^{\prime} \end{array} } \right) \]

using the angle definition presented in the following figure.

Rotation transformation and convention.

Note that, in \(\cal{R}\), the spherical angles \((\theta, \varphi)\) are not defined in the conventional way.

The rotation to move from \(\cal{R}\) to \(\cal{R}^{\prime}\) is described by the inverse of \(R\). For a rotation matrix, we simply use the transpose matrix:

\[ R^{-1} = R_z(-\alpha)\times R_x(-\beta) \times R_z(-\gamma) = R^{T} = \left[ {\begin{array}{ccc} R_{xx} & R_{yx} & R_{zx} \\ R_{xy} & R_{yy} & R_{zy} \\ R_{xz} & R_{yz} & R_{zz} \end{array} } \right] \]

Author: Florent Robinet

Constructor & Destructor Documentation

◆ Rotation()

Rotation::Rotation	(	const double	aAlpha,
		const double	aBeta,
		const double	aGamma
	)

Rotation class constuctor.

The Rotation object is initialized. The Euler angles must be provided to initialize the respective orienation of \(\cal{R}\) and \(\cal{R}^{\prime}\).

Parameters

[in]	aAlpha	Euler angle \(\alpha\) [rad].
[in]	aBeta	Euler angle \(\beta\) [rad].
[in]	aGamma	Euler angle \(\gamma\) [rad].

◆ ~Rotation()

Rotation::~Rotation ( void )

virtual

Rotation class destructor.

Member Function Documentation

◆ GetAlpha()

double Rotation::GetAlpha ( void )

inline

Returns the current value of \(\alpha\) [rad].

◆ GetBeta()

double Rotation::GetBeta ( void )

inline

Returns the current value of \(\beta\) [rad].

◆ GetCosAlpha()

double Rotation::GetCosAlpha ( void )

inline

Returns the current value of \(\cos(\alpha)\).

◆ GetCosBeta()

double Rotation::GetCosBeta ( void )

inline

Returns the current value of \(\cos(\beta)\).

◆ GetCosGamma()

double Rotation::GetCosGamma ( void )

inline

Returns the current value of \(\cos(\gamma)\).

◆ GetGamma()

double Rotation::GetGamma ( void )

inline

Returns the current value of \(\gamma\) [rad].

◆ GetOmega()

double Rotation::GetOmega ( void )

inline

Returns the current value of \(\omega\).

The angle \(\omega\) is the angle between the \(x^{\prime}\) axis and the projection of the \(x\) axis in the \(y^{\prime}x^{\prime}\) plane. In \(\cal{R}\):

\[ \vec{u}_{x} = R^{-1}\times (1,0,0) = (R_{xx}, R_{xy}, R_{xz}) = (\sin\Psi\cos\omega, \sin\Psi\sin\omega, \cos\Psi) \]

Therefore: \(\omega = \arccos(R_{xx}/\sin\Psi)\), using the sign of \(\sin\omega\) to fix \(\Psi\) in \([0,2\pi[\).

◆ GetPsi()

double Rotation::GetPsi ( void )

inline

Returns the current value of \(\Psi\).

The angle \(\Psi\) is the angle between the \(z^{\prime}\) axis and the \(x\) axis. In \(\cal{R}\):

\[ \vec{u}_{x} = R^{-1}\times (1,0,0) = (R_{xx}, R_{xy}, R_{xz}) = (\sin\Psi\cos\omega, \sin\Psi\sin\omega, \cos\Psi) \]

Therefore: \(\Psi = \arccos(R_{xz})\).

◆ GetRotationAngle()

double Rotation::GetRotationAngle ( void )

inline

Returns the rotation angle from \(\cal{R}^{\prime}\) to \(\cal{R}\) [rad].

The rotation angle \(\delta\) is obtained with:

\[ \delta = \mathrm{arccos}\left((\mathrm{trace} - 1)/2\right) \]

where \(\mathrm{trace}\) is the trace of the rotation matrix.

Note: \(0 \le \delta \le \pi\).

See also: GetRotationAxisXp(), GetRotationAxisYp(), GetRotationAxisZp() for the rotation axis.

◆ GetRotationAxisXp()

double Rotation::GetRotationAxisXp ( void )

inline

Returns the \(x^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\).

The vector \(x^{\prime}\) component is given by \(R_{zy}-R_{yz}\).

Warning: The vector is not a unit vector.

◆ GetRotationAxisYp()

double Rotation::GetRotationAxisYp ( void )

inline

Returns the \(y^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\).

The vector \(y^{\prime}\) component is given by \(R_{xz}-R_{zx}\).

Warning: The vector is not a unit vector.

◆ GetRotationAxisZp()

double Rotation::GetRotationAxisZp ( void )

inline

Returns the \(z^{\prime}\) component in \(\cal{R}^{\prime}\) of a vector defining the rotation axis from \(\cal{R}^{\prime}\) to \(\cal{R}\).

The vector \(z^{\prime}\) component is given by \(R_{yx}-R_{xy}\).

Warning: The vector is not a unit vector.

◆ GetRxx()

double Rotation::GetRxx ( )

inline

Returns the rotation matrix element \(R_{xx}\).

\[ R_{xx}=\cos\alpha\cos\gamma-\sin\alpha\cos\beta\sin\gamma \]

◆ GetRxy()

double Rotation::GetRxy ( )

inline

Returns the rotation matrix element \(R_{xy}\).

\[ R_{xy}=-\sin\alpha\cos\gamma-\cos\alpha\cos\beta\sin\gamma \]

◆ GetRxz()

double Rotation::GetRxz ( )

inline

Returns the rotation matrix element \(R_{xz}\).

\[ R_{xz}=\sin\beta\sin\gamma \]

◆ GetRyx()

double Rotation::GetRyx ( )

inline

Returns the rotation matrix element \(R_{yx}\).

\[ R_{yx}=\cos\alpha\sin\gamma+\sin\alpha\cos\beta\cos\gamma \]

◆ GetRyy()

double Rotation::GetRyy ( )

inline

Returns the rotation matrix element \(R_{yy}\).

\[ R_{yy}=-\sin\alpha\sin\gamma+\cos\alpha\cos\beta\cos\gamma \]

◆ GetRyz()

double Rotation::GetRyz ( )

inline

Returns the rotation matrix element \(R_{yz}\).

\[ R_{yz}=-\sin\beta\cos\gamma \]

◆ GetRzx()

double Rotation::GetRzx ( )

inline

Returns the rotation matrix element \(R_{zx}\).

\[ R_{zx}=\sin\alpha\sin\beta \]

◆ GetRzy()

double Rotation::GetRzy ( )

inline

Returns the rotation matrix element \(R_{zy}\).

\[ R_{zy}=\cos\alpha\sin\beta \]

◆ GetRzz()

double Rotation::GetRzz ( )

inline

Returns the rotation matrix element \(R_{zz}\).

\[ R_{zz}=\cos\beta \]

◆ GetSinAlpha()

double Rotation::GetSinAlpha ( void )

inline

Returns the current value of \(\sin(\alpha)\).

◆ GetSinBeta()

double Rotation::GetSinBeta ( void )

inline

Returns the current value of \(\sin(\beta)\).

◆ GetSinGamma()

double Rotation::GetSinGamma ( void )

inline

Returns the current value of \(\sin(\gamma)\).

◆ GetUpx()

double Rotation::GetUpx	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(u^{\prime}_x\) coordinate in the \(\cal{R}^{\prime}\) reference frame.

The coordinates in the \(\cal{R}^{\prime}\) reference frame, \((\theta^{\prime},\varphi^{\prime})\) are used to compute the x-coordinate of \(\vec{u}^{\prime}\) in the \(\cal{R}^{\prime}\) reference frame:

\[ u^{\prime}_x=\sin\theta^{\prime}\cos\varphi^{\prime} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, \(\cal{R}^{\prime}\) reference frame.
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, \(\cal{R}^{\prime}\) reference frame.

◆ GetUpy()

double Rotation::GetUpy	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(u^{\prime}_y\) coordinate in the \(\cal{R}^{\prime}\) reference frame.

The coordinates in the \(\cal{R}^{\prime}\) reference frame, \((\theta^{\prime},\varphi^{\prime})\) are used to compute the y-coordinate of \(\vec{u}^{\prime}\) in the \(\cal{R}^{\prime}\) reference frame:

\[ u^{\prime}_y=\sin\theta^{\prime}\sin\varphi^{\prime} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, \(\cal{R}^{\prime}\) reference frame.
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, \(\cal{R}^{\prime}\) reference frame.

◆ GetUpz()

double Rotation::GetUpz ( const double aThetaPrime )

inline

Returns the \(u^{\prime}_z\) coordinate in the \(\cal{R}^{\prime}\) reference frame.

The coordinates in the \(\cal{R}^{\prime}\) reference frame, \((\theta^{\prime},\varphi^{\prime})\) are used to compute the z-coordinate of \(\vec{u}^{\prime}\) in the \(\cal{R}^{\prime}\) reference frame:

\[ u^{\prime}_z=\cos\theta^{\prime} \]

Parameters

[in] aThetaPrime \(\theta^{\prime}\) coordinate, \(\cal{R}^{\prime}\) reference frame.

◆ GetUx()

double Rotation::GetUx ( const double aTheta )

inline

Returns the \(u_x\) coordinate in the \(\cal{R}\) reference frame.

The coordinates in the \(\cal{R}\) reference frame, \((\theta,\varphi)\) are used to compute the x-coordinate of \(\vec{u}\) in the \(\cal{R}\) reference frame:

\[ u_x=\cos\theta \]

Parameters

[in] aTheta \(\theta\) coordinate, \(\cal{R}\) reference frame.

◆ GetUy()

double Rotation::GetUy	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(u_y\) coordinate in the \(\cal{R}\) reference frame.

The coordinates in the \(\cal{R}\) reference frame, \((\theta,\varphi)\) are used to compute the y-coordinate of \(\vec{u}\) in the \(\cal{R}\) reference frame:

\[ u_y=\sin\theta \times \cos\varphi \]

Parameters

[in]	aTheta	\(\theta\) coordinate, \(\cal{R}\) reference frame.
[in]	aPhi	\(\varphi\) coordinate, \(\cal{R}\) reference frame.

◆ GetUz()

double Rotation::GetUz	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(u_z\) coordinate in the \(\cal{R}\) reference frame.

The coordinates in the \(\cal{R}\) reference frame, \((\theta,\varphi)\) are used to compute the z-coordinate of \(\vec{u}\) in the \(\cal{R}\) reference frame:

\[ u_z=\sin\theta \times \sin\varphi \]

Parameters

[in]	aTheta	\(\theta\) coordinate, \(\cal{R}\) reference frame.
[in]	aPhi	\(\varphi\) coordinate, \(\cal{R}\) reference frame.

◆ SetAlpha()

void Rotation::SetAlpha ( const double aAlpha = 0.0 )

inline

Sets a new \(\alpha\) value.

Parameters

[in] aAlpha New \(\alpha\) value [rad].

◆ SetAlphaBetaGamma()

void Rotation::SetAlphaBetaGamma	(	const double	aAlpha,
		const double	aBeta,
		const double	aGamma
	)

inline

Sets Euler angles, \(\alpha, \beta, \gamma\), for the rotation.

Parameters

[in]	aAlpha	Euler angle \(\alpha\) [rad].
[in]	aBeta	Euler angle \(\beta\) [rad].
[in]	aGamma	Euler angle \(\gamma\) [rad].

◆ SetBeta()

void Rotation::SetBeta ( const double aBeta = 0.0 )

inline

Sets a new \(\beta\) value.

Parameters

[in] aBeta New \(\beta\) value [rad].

◆ SetGamma()

void Rotation::SetGamma ( const double aGamma = 0.0 )

inline

Sets a new \(\gamma\) value.

Parameters

[in] aGamma New \(\gamma\) value [rad].

◆ SetUnitQuaternion()

void Rotation::SetUnitQuaternion	(	const double	aQ0,
		const double	aQ1,
		const double	aQ2,
		const double	aQ3,
		const bool	aRtoRprime = `true`
	)

Sets Euler angles from a unit quaternion.

The rotation Euler angles are derived from a unit quaternion: \(\mathbf{q} = q_0 + q_1\mathbf{i} + q_2\mathbf{j} + q_3\mathbf{k}\), where \(q_0^2+q_1^2+q_2^2+q_3^2 = 1\).

Parameters

[in]	aQ0	\(q_0\).
[in]	aQ1	\(q_1\).
[in]	aQ2	\(q_2\).
[in]	aQ3	\(q_3\).
[in]	aRtoRprime	By default, the quaternion operates the rotation from \(\cal{R}^{\prime}\) to \(\cal{R}\). Set this flag to false to define the inverse rotation from \(\cal{R}\) to \(\cal{R}^{\prime}\).

Precondition: The quaternion must be normalized. This will not be checked!

◆ TransformGetCosPhi()

double Rotation::TransformGetCosPhi	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\cos\varphi\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the cosine of the \(\varphi\) angle measured from the \(y\) axis.

\[ \cos\varphi=\frac{R_{yx}u^{\prime}_x+R_{yy}u^{\prime}_y+R_{yz}u^{\prime}_z}{\sin\theta} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).

◆ TransformGetCosPhiPrime()

double Rotation::TransformGetCosPhiPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\cos\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the cosine of the \(\varphi^{\prime}\) angle measured from the \(x^{\prime}\) axis.

\[ \cos\varphi^{\prime}=\frac{R_{xx}u_x+R_{yx}u_y+R_{zx}u_z}{\sin\theta^{\prime}} \]

Parameters

[in]	aTheta	\(\theta\) coordinate, reference frame \(\cal{R}\).
[in]	aPhi	\(\varphi\) coordinate, reference frame \(\cal{R}\).

◆ TransformGetCosTheta()

double Rotation::TransformGetCosTheta	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\cos\theta\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the cosine of the \(\theta\) angle measured from the \(x\) axis.

\[ \cos\theta=R_{xx}u^{\prime}_x+R_{xy}u^{\prime}_y+R_{xz}u^{\prime}_z \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.

◆ TransformGetCosThetaPrime()

double Rotation::TransformGetCosThetaPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\cos\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the cosine of the \(\theta^{\prime}\) angle measured from the \(z^{\prime}\) axis.

\[ \cos\theta^{\prime}=R_{xz}u_x+R_{yz}u_y+R_{zz}u_z \]

Parameters

[in]	aTheta	\(\theta\) coordinate, \(\cal{R}\) reference frame.
[in]	aPhi	\(\varphi\) coordinate, \(\cal{R}\) reference frame.

◆ TransformGetSinPhi()

double Rotation::TransformGetSinPhi	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\sin\varphi\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the sine of the \(\varphi\) angle measured from the \(y\) axis.

\[ \sin\varphi=\frac{R_{zx}u^{\prime}_x+R_{zy}u^{\prime}_y+R_{zz}u^{\prime}_z}{\sin\theta} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).

◆ TransformGetSinPhiPrime()

double Rotation::TransformGetSinPhiPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\sin\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the sine of the \(\varphi^{\prime}\) angle measured from the \(x^{\prime}\) axis.

\[ \sin\varphi^{\prime}=\frac{R_{xy}u_x+R_{yy}u_y+R_{zy}u_z}{\sin\theta^{\prime}} \]

Parameters

[in]	aTheta	\(\theta\) coordinate, reference frame \(\cal{R}\).
[in]	aPhi	\(\varphi\) coordinate, reference frame \(\cal{R}\).

◆ TransformGetSinTheta()

double Rotation::TransformGetSinTheta	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\sin\theta\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the sine of the \(\theta\) angle measured from the \(x\) axis.

\[ \sin\theta=\sqrt{\left(R_{yx}u^{\prime}_x+R_{yy}u^{\prime}_y+R_{yz}u^{\prime}_z\right)^2 +\left(R_{zx}u^{\prime}_x+R_{zy}u^{\prime}_y+R_{zz}u^{\prime}_z\right)^2} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.

◆ TransformGetSinThetaPrime()

double Rotation::TransformGetSinThetaPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\sin\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the sine of the \(\theta^{\prime}\) angle measured from the \(z^{\prime}\) axis.

\[ \sin\theta^{\prime}=\sqrt{\left(R_{xx}u_x+R_{yx}u_y+R_{zx}u_z\right)^2 +\left(R_{xy}u_x+R_{yy}u_y+R_{zy}u_z\right)^2} \]

Parameters

[in]	aTheta	\(\theta\) coordinate, \(\cal{R}\) reference frame.
[in]	aPhi	\(\varphi\) coordinate, \(\cal{R}\) reference frame.

◆ TransformGetTanPhi()

double Rotation::TransformGetTanPhi	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\tan\varphi\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the tangent of the \(\varphi\) angle measured from the \(y\) axis.

\[ \tan\varphi=\frac{R_{zx}u^{\prime}_x+R_{zy}u^{\prime}_y+R_{zz}u^{\prime}_z}{R_{yx}u^{\prime}_x+R_{yy}u^{\prime}_y+R_{yz}u^{\prime}_z} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).

◆ TransformGetTanPhiPrime()

double Rotation::TransformGetTanPhiPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\tan\varphi^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the tangent of the \(\varphi^{\prime}\) angle measured from the \(x^{\prime}\) axis.

\[ \tan\varphi^{\prime}=\frac{R_{xy}u_x+R_{yy}u_y+R_{zy}u_z}{R_{xx}u_x+R_{yx}u_y+R_{zx}u_z} \]

Parameters

[in]	aTheta	\(\theta\) coordinate, reference frame \(\cal{R}\).
[in]	aPhi	\(\varphi\) coordinate, reference frame \(\cal{R}\).

◆ TransformGetTanTheta()

double Rotation::TransformGetTanTheta	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\tan\theta\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns the tangent of the \(\theta\) angle measured from the \(x\) axis. It is obtained from TransformGetSinTheta() and TransformGetCosTheta() functions.

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, \(\cal{R^{\prime}}\) reference frame.

◆ TransformGetTanThetaCosPhi()

double Rotation::TransformGetTanThetaCosPhi	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\tan\theta\cos\varphi\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns \(\tan\theta\cos\varphi\) where the \(\varphi\) angle is measured from the \(y\) axis and the \(\theta\) angle is measured from the \(x\) axis.

\[ \tan\theta\cos\varphi=\frac{R_{yx}u^{\prime}_x+R_{yy}u^{\prime}_y+R_{yz}u^{\prime}_z}{R_{xx}u^{\prime}_x+R_{xy}u^{\prime}_y+R_{xz}u^{\prime}_z} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, reference frame \(\cal{R}\).
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).

◆ TransformGetTanThetaPrime()

double Rotation::TransformGetTanThetaPrime	(	const double	aTheta,
		const double	aPhi
	)

inline

Returns the \(\tan\theta^{\prime}\) in the reference frame \(\cal{R}^{\prime}\).

The coordinates in the reference frame \(\cal{R}\), \((\theta,\varphi)\) are transformed into coordinates in the reference frame \(\cal{R}^{\prime}\). This function returns the tangent of the \(\theta^{\prime}\) angle measured from the \(z^{\prime}\) axis. It is obtained from TransformGetSinThetaPrime() and TransformGetCosThetaPrime() functions.

Parameters

[in]	aTheta	\(\theta\) coordinate, \(\cal{R}\) reference frame.
[in]	aPhi	\(\varphi\) coordinate, \(\cal{R}\) reference frame.

◆ TransformGetTanThetaSinPhi()

double Rotation::TransformGetTanThetaSinPhi	(	const double	aThetaPrime,
		const double	aPhiPrime
	)

inline

Returns the \(\tan\theta\sin\varphi\) in the reference frame \(\cal{R}\).

The coordinates in the reference frame \(\cal{R^{\prime}}\), \((\theta^{\prime},\varphi^{\prime})\) are transformed into coordinates in the reference frame \(\cal{R}\). This function returns \(\tan\theta\sin\varphi\) where the \(\varphi\) angle is measured from the \(y\) axis and the \(\theta\) angle is measured from the \(x\) axis.

\[ \tan\theta\sin\varphi=\frac{R_{zx}u^{\prime}_x+R_{zy}u^{\prime}_y+R_{zz}u^{\prime}_z}{R_{xx}u^{\prime}_x+R_{xy}u^{\prime}_y+R_{xz}u^{\prime}_z} \]

Parameters

[in]	aThetaPrime	\(\theta^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).
[in]	aPhiPrime	\(\varphi^{\prime}\) coordinate, reference frame \(\cal{R^{\prime}}\).

Member Data Documentation

◆ c_alpha

double Rotation::c_alpha

private

\(\cos(\alpha)\).

◆ c_beta

double Rotation::c_beta

private

\(\cos(\beta)\).

◆ c_gamma

double Rotation::c_gamma

private

\(\cos(\gamma)\).

◆ rot_alpha

double Rotation::rot_alpha

protected

Euler angle \(\alpha\).

◆ rot_beta

double Rotation::rot_beta

protected

Euler angle \(\beta\).

◆ rot_gamma

double Rotation::rot_gamma

protected

Euler angle \(\gamma\).

◆ s_alpha

double Rotation::s_alpha

private

\(\sin(\alpha)\).

◆ s_beta

double Rotation::s_beta

private

\(\sin(\beta)\).

◆ s_gamma

double Rotation::s_gamma

private

\(\sin(\gamma)\).

The documentation for this class was generated from the following files:

src/Geometry/Rotation.h
src/Geometry/Rotation.cc

Public Member Functions

Protected Attributes

Private Attributes

Detailed Description

Defintions

Change of reference frames

Constructor & Destructor Documentation

◆ Rotation()

◆ ~Rotation()

Member Function Documentation

◆ GetAlpha()

◆ GetBeta()

◆ GetCosAlpha()

◆ GetCosBeta()

◆ GetCosGamma()

◆ GetGamma()

◆ GetOmega()

◆ GetPsi()

◆ GetRotationAngle()

◆ GetRotationAxisXp()

◆ GetRotationAxisYp()

◆ GetRotationAxisZp()

◆ GetRxx()

◆ GetRxy()

◆ GetRxz()

◆ GetRyx()

◆ GetRyy()

◆ GetRyz()

◆ GetRzx()

◆ GetRzy()

◆ GetRzz()

◆ GetSinAlpha()

◆ GetSinBeta()

◆ GetSinGamma()

◆ GetUpx()

◆ GetUpy()

◆ GetUpz()

◆ GetUx()

◆ GetUy()

◆ GetUz()

◆ SetAlpha()

◆ SetAlphaBetaGamma()

◆ SetBeta()

◆ SetGamma()

◆ SetUnitQuaternion()

◆ TransformGetCosPhi()

◆ TransformGetCosPhiPrime()

◆ TransformGetCosTheta()

◆ TransformGetCosThetaPrime()

◆ TransformGetSinPhi()

◆ TransformGetSinPhiPrime()

◆ TransformGetSinTheta()

◆ TransformGetSinThetaPrime()

◆ TransformGetTanPhi()

◆ TransformGetTanPhiPrime()

◆ TransformGetTanTheta()

◆ TransformGetTanThetaCosPhi()

◆ TransformGetTanThetaPrime()

◆ TransformGetTanThetaSinPhi()

Member Data Documentation

◆ c_alpha

◆ c_beta

◆ c_gamma

◆ rot_alpha

◆ rot_beta

◆ rot_gamma

◆ s_alpha

◆ s_beta

◆ s_gamma